0. Introduction: In the last twenty years, several remarkable

results have been established in the context of infinite dimensional

Banach space theory. Most of these results emphasize the interplay

between the topological, geometrical and measure theoretical

structures of a Banach space. Here are two well known prototypes of

such interrelations. They are due to the combined efforts of several

authors and we refer the reader to the books [8] and [14] for a

detailed account of their history and for the notions involved in

their statements.

Theorem (A): Let X be a Banach space and let D be a closed convex

bounded subset of X. The following properties are then equivalent:

(i) All D-valued martingales norm converge almost surely,

(ii) Every non-empty subset of D has slices of arbitrarily small

diameter.

Theorem (B): Let Y be a separable Banach space and let D be a

* *

w -compact convex subset of Y • The following properties are then

equivalent:

(i) All D-valued martingales converge in the Pettis-norm.

**

(ii) Every functional in Y is the pointwise limit on D of a

bounded sequence of elements in Y.

A set verifying the conditions of Theorem (A) is then said to

have the Radon-Nikodym property (R.N.P) while a set verifying those of

Theorem (B) is said to have the Weak Radon-Nikodym property

(W.R.N.P). In both cases, the set is then the norm closed convex hull

of its extreme points.

The concept of a Radon-Nikodym set turned out to be central in

the study of extremal structures in convex sets, integral

representations and problems involving non-linear optimization ([8],

[14], [19]). On the other hand, the weak Radon-Nikodym property (for

w*-compact sets) is closely related to the classical Baire theory of

functions and its relatively recent resurgence with the deep theorems

of Bourgain-Fremlin-Talagrand (B.F.T [5]) following the pioneering

work of Rosenthal [38] and Odell-Rosenthal [33].

However, even though these two concepts are now well understood

Received by the editors September 26, 1986.

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